Question

Write cos(sin^-1(-x/4)) as an algebraic expression in x

Answer 1

Let \(y=\sin^{-1}\left(-\dfrac{x}{4}\right)\), then \(\sin y=-\dfrac{x}{4}\). If you draw a reference triangle, you have

There are two possible values for \(\theta\) here. However, the domain of the inverse sine function is restricted to the first and second/fourth quadrants of the unit circle. We don't have a triangle that fits the description drawn in the first or second quadrants, so we use the one in the fourth. The missing (horizontal) side:
\[(-x)^2+\Box^2=4^2~~\Rightarrow~~\Box=\sqrt{16-x^2}\]
You take the positive root because the triangle has a positive horizontal component.

From this, you can find the cosine:
\[\cos\left(\sin^{-1}\left(-\frac{x}{4}\right)\right)=\cos\theta=\frac{\sqrt{16-x^2}}{4}\]